60.2.91 problem 667

Internal problem ID [10678]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 667
Date solved : Monday, January 27, 2025 at 09:25:06 PM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {y^{3} {\mathrm e}^{-2 b x}}{y \,{\mathrm e}^{-b x}+1} \end{align*}

✓ Solution by Maple

Time used: 0.382 (sec). Leaf size: 94

dsolve(diff(y(x),x) = y(x)^3/(y(x)*exp(-b*x)+1)*exp(-2*b*x),y(x), singsol=all)
 

\[ -\frac {-\ln \left (y \,{\mathrm e}^{-b x}\right ) \sqrt {b \left (4+b \right )}+\left (\frac {\ln \left (-b y \,{\mathrm e}^{-b x}+y^{2} {\mathrm e}^{-2 b x}-b \right )}{2}-b x +c_{1} \right ) \sqrt {b \left (4+b \right )}+b \,\operatorname {arctanh}\left (\frac {-2 y \,{\mathrm e}^{-b x}+b}{\sqrt {b \left (4+b \right )}}\right )}{\sqrt {b \left (4+b \right )}} = 0 \]

✓ Solution by Mathematica

Time used: 0.526 (sec). Leaf size: 130

DSolve[D[y[x],x] == y[x]^3/(E^(2*b*x)*(1 + y[x]/E^(b*x))),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{-\frac {e^{2 b x} \left (e^{b x} b+(b+3) y(x)\right )}{\sqrt [3]{-b^2 (2 b+9) e^{6 b x}} \left (y(x)+e^{b x}\right )}}\frac {1}{K[1]^3+\frac {3 \sqrt [3]{-1} (b+3) K[1]}{\sqrt [3]{b} (2 b+9)^{2/3}}+1}dK[1]=\frac {1}{9} x e^{-4 b x} \left (-b^2 (2 b+9) e^{6 b x}\right )^{2/3}+c_1,y(x)\right ] \]